TL;DR
This paper discusses the role of mod-$\,\ell$ cohomology of Deligne--Lusztig varieties in understanding the modular representation theory of finite reductive groups, based on lecture notes from 2016.
Contribution
It provides an exposition highlighting the significance of mod-$\ell$ cohomology in modular representation theory, with insights from recent lectures.
Findings
Emphasizes the importance of mod-$\ell$ cohomology in modular representation theory.
Connects geometric methods to algebraic representations of finite groups.
Provides a pedagogical overview based on recent lectures.
Abstract
These notes are based on a series of lectures given by the author at the Centre Bernoulli (EPFL) in July 2016. They aim at illustrating the importance of the mod- cohomology of Deligne--Lusztig varieties in the modular representation theory of finite reductive groups.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Lectures on modular Deligne–Lusztig theory
Olivier Dudas
(Date: CIB, July 2016)
Abstract.
These notes are based on a series of lectures given by the author at the Centre Bernoulli (EPFL) in July 2016. They aim at illustrating the importance of the mod- cohomology of Deligne–Lusztig varieties in the modular representation theory of finite reductive groups.
Introduction
In order to construct and study the complex representations of finite reductive groups (such as , ,…) Deligne and Lusztig introduced in 1976 a family of algebraic varieties acted on by [20]. The subsequent work of Lusztig on the cohomology of these Deligne–Lusztig varieties led to a complete classification of the irreducible characters of finite reductive groups [53].
The purpose of these lectures is to present a generalization of the theory of Deligne–Lusztig to the modular setting, that is, for representations over fields of positive characteristic. This originated in the work of Broué [11] and Bonnafé–Rouquier [5].
In the ordinary case (in characteristic zero), the representation theory is controlled by the simple objects, which are in turn determined by a numerical datum, their characters. The situation is far more complicated for representations in positive characteristic; several classes of indecomposable objects are of particular interest, and more information is needed to understand the representations, namely:
- •
information of numerical nature: characters of projective modules, multiplicities of simple modules in a given ordinary character, all of which are encoded in the so-called decomposition matrix;
- •
information of homological nature: extensions between simple modules, Loewy series of projective modules, projective resolutions of simple objects.
The alternating sum of the cohomology groups of Deligne-Lusztig varieties produces a virtual character – an element of the Grothendieck group of the category of representations. In the modular framework, this object does not contain enough information, and one should consider each individual cohomology group, or rather the cohomology complex of the variety. This object now lives in the derived category of representations, and it encodes many aspects of the modular representation theory of the group. One crucial incarnation of this phenomenon is the geometric version of Broué’s abelian defect group conjecture, which predicts that the cohomology complex of a suitably chosen Deligne–Lusztig variety induces a derived equivalence between the principal block of a finite reductive group and its Brauer correspondent. Not so many cases of this conjecture are known to hold but a lot of numerical evidence and many partial results have been obtained in that direction.
The first part of these lectures aims at introducing the mod- cohomology of Deligne–Lusztig varieties using the modern language of derived and homotopy categories. Unlike most of the textbooks on étale and -adic cohomology, we avoid the definition and focus on the properties of the cohomology complexes of varieties acted on by a finite group (such as perfectness), with particular attention on how one can compute such complexes (using decompositions, quotients or fixed points). In the second part we present several recent results obtained using this approach. They include the computation of decomposition numbers in §4 (a joint work with G. Malle) and the determination of Brauer trees in §5 (a joint work with D. Craven and R. Rouquier). This illustrates how powerful the geometric methods are for solving representation theoretic problems for finite reductive groups. There is a converse to that statement, and we explain in a final chapter how to use representation theory to show that the cohomology of a particular Deligne–Lusztig variety is torsion-free.
Contents
1. Introduction to derived categories
Throughout this chapter, will denote a ring with unit. The category of left -modules (resp. finitely generated left -modules) will be denoted by (resp. ).
The purpose of this first chapter is to introduce two categories, the homotopy category and the derived category . Here is a non-exhaustive list of reasons why we are going to work in this framework, instead of working with -modules or complexes of -modules:
- •
to get rid of (split) exact sequences;
- •
to have uniqueness of projective or injective resolutions;
- •
to have a good notion of duality (e.g. over );
- •
to work with non-exact functors.
Several steps are needed to understand the construction of the homotopy and derived categories of -modules:
[TABLE]
getting rid of split
exact sequences
inverting
quasi-isomorphisms
Note however that this chapter is not intended to provide a detailed account on this construction. For further reading on the subject we recommend for example the excellent textbooks by Gelfand–Manin [42] and Neeman [55].
1.1. Complexes of -modules
A complex of -modules is
[TABLE]
where, for each , is a left -module and is a morphism of -modules (the differential) satisfying .
A morphism between two complexes is given by a family of morphisms of -modules making the following diagram commute
[TABLE]
Here is an example of a morphism between two complexes of -modules (we will see later that it is a quasi-isomorphism)
[TABLE]
The category of complexes of -modules will be denoted . It is an abelian category. This can be seen by considering the ring with . Then the functor
[TABLE]
is an equivalence of categories. The abelian structure on is obtained via the equivalence from the abelian structure of . As a consequence, we can consider kernels and cokernels of morphisms between complexes, as well as exact sequences of complexes.
We say that a complex is bounded above (resp. bounded below, resp. bounded) if for (resp. , resp. ). The corresponding full subcategory of will be denoted by (resp. , resp. ).
Given and a complex of -modules, we define the -th shift of , denoted by , to be the complex with terms and differential . If is an -module, the notation stands for the complex with zero terms outside the degree and in the degree . The functor
[TABLE]
is fully faithful. In other words, can be identified with complexes with zero terms outside the degree [math].
Since , is a submodule of . The quotient is an -module, called the degree cohomology group of . We write
[TABLE]
We say that a complex is exact or acyclic if for all . The compatibility of maps between complexes and the respective differentials ensures that any morphism of complexes induces a family of morphisms of -modules .
From now on, we will omit the subscript ∙ in the notation of complexes, as well as the reference to the differentials for morphisms of complexes.
Proposition 1.1**.**
Let be a short exact sequence of complexes of -modules. Then there are boundary maps for all yielding a long exact sequence of -modules
[TABLE]
Sketch of proof.
Let , which we write for some . Since is a morphism of complexes, , hence . Now write and set . ∎
Exercise 1.2*.*
Check that is well-defined, and that it induces the long exact sequence stated in the proposition.
In terms of -modules, the proposition shows the existence of a morphism of -modules which fits in a triangle
[TABLE]
1.2. The homotopy category
A morphism between complexes of -modules is said to be null-homotopic if it is of the form for some map (not necessarily a morphism of complexes). We illustrate this with the following diagram:
[TABLE]
Each vertical map satisfies
Given two morphisms of complexes we write if is null-homotopic. This is an equivalence relation, compatible with the sum and the composition of morphisms. A complex is null-homotopic if the identity map is null-homotopic (i.e. ). We say that is a homotopy equivalence if there exists a morphism such that and .
Definition 1.3**.**
The homotopy category of -modules, denoted by , is the category with
- •
objects: complexes of -modules (same as ),
- •
morphisms: .
It is an additive category (but non-abelian in general). The isomorphisms in the homotopy category are exactly the classes of the homotopy equivalences.
Exercise 1.4*.*
Let be a short exact sequence of -modules, and be the complex associated to this sequence, with in degree [math]. Show that is null-homotopic if and only if the exact sequence splits.
A complex of the form is null-homotopic if and only if is an isomorphism. More generally, a complex is null-homotopic if and only if it decomposes as a direct sum of complexes of the form
[TABLE]
See also Exercise 1.4. When working in we will often consider reduced complexes where all the null-homotopic direct summands are removed.
When is null-homotopic, the corresponding morphism on cohomology groups is zero. As a consequence, if then . Also, if is a homotopy equivalence then is an isomorphism. In particular, if in then .
We mentioned earlier that projective and injective resolutions are unique in the homotopy and derived categories. Recall that a projective resolution of an -module is a complex of projective -modules
[TABLE]
such that . In other words, there is a surjective map which fits in the following long exact sequence
[TABLE]
The idea of projective resolutions is to replace by a complex with the same cohomology ( in degree [math]) but whose terms are ‘nicer’.
Proposition 1.5**.**
If and are two projective resolutions of then in .
Sketch of proof.
We only show how to construct the morphisms and which will be mutually inverse in the homotopy category.
Let us denote by (resp. ) the differential of the complex (resp. ) and by (resp. ) the respective projections. Since is surjective and is projective, the map factors through . In other words, there exists such that .
Since , we have . Therefore can be seen as a map from to . Since is projective, it should factor through the surjective map . In other words, there exists such that . By iterating this construction, we obtain, for all , maps such that . This means that is a morphism of complexes. The construction of is similar. ∎
Exercise 1.6*.*
Show that and .
More generaly, if is a complex of -modules, a projective resolution of is a bounded above complex of projective modules together with a morphism such that is an isomorphism for all (a quasi-isomorphism, see §1.3). We shall see in Proposition 1.15 that is uniquely determined by up to homotopy equivalence, which generalizes the previous proposition.
Exercise 1.7*.*
Spell out the case where has only one non-zero term, say in degree [math].
Definition 1.8**.**
The mapping cone of a morphism of complexes is the complex with differential
[TABLE]
If , the mapping cone is just the direct sum of and in the category of complexes. However, if is non-zero, it encodes more information. For example, if and are complexes concentrated in degree [math], then is induced by a morphism of -modules . In this case the complex has only two non-zero terms (in degree and [math]) and its cohomology is and . We deduce the existence of a long exact sequence of -modules
[TABLE]
This generalizes to morphisms of complexes as follows.
Proposition 1.9**.**
Any morphism of complexes induces a long exact sequence
[TABLE]
The maps and are induced by the natural morphisms of complexes and . Note that . The maps and are non-zero in general, and only null-homotopic. This ensures that we have a triangle
[TABLE]
in the homotopy category .
Exercise 1.10*.*
Show that and are null-homotopic.
More generally, a triangle in is such that , and are null-homotopic. We represent it as or as a triangular diagram
[TABLE]
A morphism of triangles between and is the data of morphisms of complexes , and making the following diagram commute
[TABLE]
A triangle is distinguished if it is isomorphic to a triangle for some morphism of complexes . In particular, by Proposition 1.9 any distinguished triangle yields a long exact sequence in cohomology
[TABLE]
The category together with the suspension and the collection of distinguished triangles is a triangulated category (see for example [55, Chap. 1] for the list of axioms of triangulated categories).
Morphisms in the homotopy category can be expressed in terms of the cohomology of the total Hom complex. Given and two complexes of -modules, the total Hom complex, denoted by , is defined by
[TABLE]
with the differential given by
[TABLE]
for every . One can readily check that consists of the morphisms of complexes from to whereas is the subgroup of null-homotopic morphisms. Consequently,
[TABLE]
1.3. The derived category of -
A morphism of complexes is a quasi-isomorphism if the maps induced on the cohomology groups are isomorphisms for all . By Proposition 1.9 this is equivalent to the complex being acyclic.
The derived category is obtained from the homotopy category by formally inverting the quasi-isomorphisms (equivalently, by taking the quotient by the acyclic complexes). This is analogous to the construction of the fraction field of a domain. More precisely, given two pairs of morphisms in , say and with and being quasi-isomorphisms, we write if there exists a commutative diagram in
[TABLE]
with a quasi-isomorphism. This represents the relation “”.
Definition 1.11**.**
The derived category of -modules, denoted by , is the category with
- •
objects: complexes of -modules (same as and ),
- •
morphisms:
[TABLE]
The natural functor (sending a morphism to the class of ) sends quasi-isomophisms to isomorphisms. It is universal for this property. A distinguished triangle in will be by definition the image of a distinguished triangle in . This endows the derived category with a structure of triangulated category. We can think of distinguished triangles in as analogues of short exact sequences of complexes.
Proposition 1.12**.**
The functor is well-defined on .
Any short exact sequence of complexes in yields a distinguished triangle in .
The functor sending a module to the complex is fully faithful.
Note however that in general does not imply in . For example and are not isomorphic in .
1.4. Morphisms in
By definition, morphisms in the derived category are equivalence classes of pairs of morphisms in the homotopy category , representing “”. We explain here how replacing and by projective or injective resolutions helps finding nice representatives for these morphisms.
We will denote by the full subcategory of whose objects are the projective -modules. The corresponding categories and correspond to the full subcategories of and respectively, whose objets are complexes of projective -modules. Similarly, will refer to the additive category of injective -modules, and (resp. ) to the corresponding category of complexes (resp. the homotopy category).
Lemma 1.13**.**
Let be a bounded above complex of projective -modules, and be two complexes of -modules.
If is an acyclic complex, any morphism is null-homotopic.
Any quasi-isomorphism splits in .
Proof.
Let be an acyclic complex and be a morphism. We can assume without loss of generality that for . We construct a homotopy as follows:
[TABLE]
Since is a morphism of complexes, then hence . Therefore since is projective there exists such that .
Similarly, using the fact that , there exists such that . We iterate this construction to get (i).
For (ii) we consider the distinguished triangle . By (i) the morphism is null-homotopic, yielding a map . If denotes the second projection of this map then one can show that is a morphism of complexes and . ∎
Exercise 1.14*.*
Check that is a morphism of complexes satisfying .
Proposition 1.15**.**
Given and there is a natural isomorphism of -modules
[TABLE]
Proof.
For the injectivity of the map, let be such that . This means that there is a commutative diagram
[TABLE]
with a quasi-isomorphism. By Lemma 1.13, there exists such that . From we get in .
The surjectivity is another application of Lemma 1.13. Indeed, any pair of morphisms can be completed in a commutative diagram
[TABLE]
with satisfying . ∎
For we will denote by and the essential images of the categories of bounded/bounded below/bounded above complexes of -modules. The following theorem follows from Proposition 1.15 and the existence of projective (or injective) resolutions. The proof is left as an exercise.
Theorem 1.16**.**
The functor induces equivalences
[TABLE]
Remark 1.17*.*
More generally, if one works with another abelian category instead of , then the first isomorphism in Theorem 1.16 (resp. the second isomorphism) remains true if has enough projective objects (resp. enough injective objects). In these notes we shall often use that , the category of finitely generated -modules has enough projectives. This guarantees the existence of projective resolutions of bounded above complexes.
1.5. Derived functors
Let be another ring with unit. Given an additive functor we can form the triangulated functors
[TABLE]
where the first equivalences are quasi-inverses of the ones given in Theorem 1.16.
When is right exact and , is a complex whose terms in positive degrees are zero and which satisfies . In that case we refer to as the left derived functor of . Similarly, when is left exact, is the right derived functor and it satisfies for every -module . Historically, only the lower (resp. higher) cohomology groups of (resp. ) were considered, not the complex itself. They yield additive functors between the module categories which we will denote by and . Note that if is exact then .
Example 1.18*.*
(a) Given an -module , the functor is an additive, covariant, left exact functor from to . It yields a right derived functor
[TABLE]
Given another -module , the group of degree extensions between and is by definition . Therefore we have
[TABLE]
Note that more generally, the definition of makes sense whenever is a bounded above complex of -modules.
(b) Given a right -module , the functor is a right exact functor (not exact if is not flat). The corresponding left derived functor is denoted by
[TABLE]
It is defined by where is any projective resolution of . This left derived functor yields the torsion groups for any non-negative integer .
1.6. Truncation and applications
Given a complex of -modules , one can consider the following truncations of :
[TABLE]
The truncated complexes are constructed so that they have the same cohomology up to (or starting from) a given degree. For example,
[TABLE]
The following proposition summarizes the relations and properties of truncation operations.
Proposition 1.19**.**
Let be a complex of -modules.
The natural map is a quasi-isomorphism.
There are short exact sequences of complexes
[TABLE]
The truncation operations are functorial. They preserve the class of acyclic complexes, null-homotopic complexes, null-homotopic morphisms and quasi-isomorphisms.
Note that similar statements are obtained by reversing the arrows and swapping and with and respectively.
We deduce that the truncation functors induce functors at the level of the homotopy and derived categories. The short exact sequences in (ii) together with the quasi-isomorphism in (i) yield distinguished triangles in
[TABLE]
Consequently, a complex whose homology vanishes outside the degrees is quasi-isomorphic to its truncation , hence to a bounded complex whose terms are zero outside the degrees . In particular a complex with a unique non-zero homology group is quasi-isomorphic to a module shifted in that degree. Another consequence is that the category is the full subcategory of with objects satisfying for .
Another example that will often appear in these notes is the case of a complex with only two non-zero homology groups, say and . Then fits into a distinguished triangle
[TABLE]
which means that is quasi-isomorphic to the cone of the map . This implies that is determined by , and by an element of (see Example 1.18).
1.7. Examples of derived categories
When is a semisimple algebra, every injective or surjective map between modules splits. Consequently one can easily show that every complex of -modules is quasi-isomorphic to the complex formed by its cohomology groups (with zero differential). In other words, the functor induces an equivalence between and the category of -graded modules. This is in particular the case for the derived category of -vector spaces when is a field, or more generally for the derived category of -modules when is a finite group whose order is invertible in .
A ring is said to be hereditary if for all . In that case, every bounded complex is again quasi-isomorphic to its cohomology, but not in a canonical way, and the functor is not faithful in general. This is for example the case for the bounded derived category of abelian groups . Another example is the bounded derived category of -modules when is a finite group whose order is prime to .
Exercise 1.20*.*
When is hereditary, show using (1.2) and Example 1.18.a that every bounded complex is quasi-isomorphic to its cohomology.
1.8. The stable category
Let be a field. Throughout this section we will assume that is a finite dimensional -algebra. In particular every finite dimensional -module has a projective cover, which we will denote by ( is said to be semiperfect). We shall also assume that is symmetric (i.e. is isomorphic to its dual as an -bimodule). In that case -modules are projective if and only if they are injective (see for example [2, §1.6]). Consequently any finite dimensional -module admits an injective resolution. A typical example of such an algebra in these lectures is the group algebra of a finite group .
Definition 1.21**.**
The stable category of finitely generated -modules, denoted by , is the category with
- •
objects: finitely generated -modules (same as ),
- •
morphisms: where if and only if factors through a projective module.
In particular, in the stable category any projective module is isomorphic to zero. There is a canonical additive (in fact -linear) functor , making into an additive (-linear) category. This category has an additional triangulated structure, as we will see below.
Given a finite dimensional -module , we define the Heller operator by
[TABLE]
We then define inductively for with the convention that is the minimal submodule of such that is projective.
Exercise 1.22*.*
Check that is well defined up to isomorphism. Show that is functorial in (but not in ).
Similarly, we set where is an injective hull of . One can readily check that as right -modules and more generally that for all .
Proposition 1.23**.**
Let and be finitely generated -modules.
* in if and only if there exist finitely generated projective modules and such that in .*
If and are indecomposable non-projective modules with or being simple, then .
If then
[TABLE]
Proof.
(i) If in then there exists morphisms and projective modules and such that and factor through and respectively. Write with and and consider the following morphisms
[TABLE]
which satisfy . This shows that is a direct summand of . Similarly, is a direct summand of and we can invoke Krull-Schmidt Theorem to conclude.
For (ii) it is enough to see that no injective (resp. surjective) morphism can factor through a projective (hence injective) module if both and have no non-trivial projective summands.
For (iii) we start with a minimal projective resolution of , given by
[TABLE]
By definition (see Example 1.18), the degree extension group is the quotient of the subgroup of maps in such that by the maps which factor through . In particular we have a well defined surjective map
[TABLE]
Now if there exists a projective (hence injective) module such that factors through . Since is injective and is a submodule of , the map can be extended to a map in the following commutative diagram
[TABLE]
Therefore factors through and hence it is zero in . The case of is similar. ∎
Consequently, any short exact sequence in yields an exact sequence in . This, in turn, endows with a structure of triangulated category, with suspension functor (or shift) , such that the images of short exact sequences of are distinguished triangles in .
The triangulated structure appears in a more natural way from the bounded derived category of . We say that a complex of -modules is perfect if it is quasi-isomorphic to a bounded complex of finitely generated projective modules. We denote by the full subcategory of of perfect complexes (it is a thick subcategory, i.e. stable under direct summands and cones).
Theorem 1.24** (Rickard [56]).**
The natural functor induces an equivalence of triangulated categories
[TABLE]
inverting maps whose cone
is a perfect complex
Remark 1.25*.*
Let us consider a projective resolution of , truncated in degrees above zero
[TABLE]
Then in . On the other hand, since all the terms of are projective modules except the term in degree we have in . This shows that
[TABLE]
and proves the compatibility of the suspension functors ( and ) under the equivalence given by Rickard’s theorem.
2. Varieties and cohomology
The aim of this chapter is to introduce the geometric tools that we will need to construct the representations of finite reductive groups. They will be obtained from linear invariants (cohomology groups or cohomology complexes) of algebraic varieties acted on by finite groups:
[TABLE]
For example, if is a finite set acted on by , then we can form the permutation module over any ring .
Although this construction makes sense for any abstract finite group , it will be particularly suited for finite reductive groups, since in that case the algebraic variety will be constructed from the underlying algebraic group (see §3 for the definition of Deligne–Lusztig varieties).
Since we will be interested in modular representations (with coefficients in fields of positive characteristic) the language of derived categories and derived functors introduced in the previous chapter will be particularly suited for our purpose:
-modules ,
\ \ \rightsquigarrow\ \
a projective resolution of
\ \ \rightsquigarrow\ \
\left\{\hskip-3.69885pt\begin{array}[]{l}P\otimes_{A}N\\ \mathrm{Hom}_{A}^{\bullet}(P,N)\end{array}\right.
\ \ \rightsquigarrow\ \
\left\{\hskip-3.69885pt\begin{array}[]{l}\mathrm{Tor}_{i}^{A}(M,N)\\ \mathrm{Ext}_{A}^{i}(M,N)\end{array}\right.
a sheaf on
\ \ \rightsquigarrow\ \
a flabby resolution of
\ \ \rightsquigarrow\ \
\ \ \rightsquigarrow\ \
derived
setting
standard
setting
The definition of étale or -adic cohomology would go far beyond the scope of these notes. For the reader interested in the topic we recommend reading Deligne’s notes [19], or the excellent textbook by Milne [54]. For a more representation-theoretic perspective, most of the properties listed in this chapter are also addressed in [15, Appendix A3] and in [3, Appendix A].
2.1. Definition and first properties
Let be a quasi-projective variety over and be a finite group acting on . We fix a prime number and an -modular system such that is a finite extension of . Finally, we denote by any ring among , and . We will be interested in representations of over .
The theory of étale cohomology of sheaves on produces two complexes of -modules and , unique up to quasi-isomorphism, called the cohomology complex of and the cohomology complex with compact support of . By extension of scalars, we also have complexes
[TABLE]
When (resp. ) we will refer to these complexes as the -adic cohomology complexes (resp. the mod- cohomology complexes). The groups
[TABLE]
are the cohomology groups (or cohomology groups with compact support) with coefficients in .
The cohomology is functorial: if is a -equivariant morphism of algebraic varieties then it induces a morphism in
[TABLE]
between the cohomology complexes of and . If in addition is proper (e.g. is a finite morphism), the same holds for the cohomology complexes with compact support.
The cohomology complexes are “small”: the -modules and are finitely generated over . Moreover, they vanish for and . Consequently, and are quasi-isomorphic to complexes of (-free) finitely generated -modules with terms in degrees .
Remark 2.1*.*
The -modules are not free in general, but is. If is a -free resolution of a given representative of , then is quasi-isomorphic to and the terms of the latter complex are -free since is (see also Proposition 6.5).
The following theorem, due to Rickard [57] (see also [59]), gives the most satisfactory representative for the cohomology complex of from a representation-theoretic perspective.
Theorem 2.2** (Rickard [57]).**
* and are quasi-isomorphic to bounded complexes whose terms are direct summands of finite sums of permutation modules for .*
Idea of proof.
Given a sheaf on , we can construct the complex from the global sections of the Godement resolution. This resolution involves -modules of the form on which acts naturally. This suggests that has a representative in the category generated by the permutation modules and we can invoke the finiteness property of the cohomology groups to conclude. ∎
For or , we denote by the category of finitely generated -permutation -modules. This is the smallest full subcategory of closed under direct summands and containing the permutation modules. As a consequence of Theorem 2.2, there exist (unique up to isomorphism) bounded complexes and in of finitely generated -permutation modules which are quasi-isomorphic to and respectively. The following particular case will be intensively used in these notes.
Corollary 2.3**.**
Assume that for all the order of the group is invertible in . Then and are perfect complexes.
For affine varieties, the vanishing property of cohomology groups can be refined. If is an affine variety of pure dimension (i.e. all the irreducible components have the same dimension) then
if ;
if .
Consequently, (resp. ) has a representative with terms in degrees (resp. ).
We conclude this section by the relation between the compact and non-compact versions of the cohomology complexes. There is a natural map which is an isomorphism when is a projective variety (compare with the case of affine varieties above). In addition, the cohomology complexes of smooth varieties are mutually dual.
Theorem 2.4** (Poincaré-Verdier [19]).**
Assume that is smooth of pure dimension . Then
[TABLE]
*not necessary to right derive if one works
with a representative of
with -free terms*
in .
2.2. Tools for computing
Unless otherwise stated, all the isomorphisms considered in this section are in the category for a ring among , and .
Theorem 2.5** (Künneth formula).**
The cohomology of a product of varieties is given by
[TABLE]
Theorem 2.6** (Open-closed situation).**
Let be an open -stable subvariety of , and be the closed complement. There is a distinguished triangle
[TABLE]
in which splits if is also closed.
Taking the cohomology of this distinguished triangle yields a long exact sequence of cohomology groups
[TABLE]
Theorem 2.7**.**
The cohomology of the affine space of dimension is given by
[TABLE]
These three results are enough to compute the cohomology of a large class of varieties. We give below the examples of projective spaces and tori.
Example 2.8*.*
(a) Write . Theorems 2.6 and 2.7 yield a distinguished triangle
[TABLE]
zero map since
This shows that .
More generally, . This method works for any variety paved by affine spaces, e.g. the flag varieties.
(b) Let be the one-dimensional torus acted on by multiplication by the group of -th roots of unity in . Again, we have a distinguished triangle
[TABLE]
element of
The long exact sequence in cohomology gives but we need more information to compute the cohomology complex.
Since acts freely, the complex is perfect by Corollary 2.3. Therefore it is quasi-isomorphic to with and being two finitely-generated projective modules in degrees and respectively. From the previous computation we deduce that . Consequently, the trivial module is in the head of and the projective cover of is a direct summand of . In other words, has a representative of the form
[TABLE]
with . This implies that the composition is surjective. Since is projective, it must split and we can write such that the restriction of to is the identity. This shows that the previous complex is homotopy equivalent to
[TABLE]
To determine we can either use the kernel of or the fact that in the Grothendieck group which forces . Finally,
[TABLE]
Using Theorem 2.5 we can also compute the cohomology of a higher dimensional torus by .
Exercise 2.9*.*
Let be a primitive -th root of . Show that
[TABLE]
2.3. Group action
Recall that is a finite group acting on the quasi-projective variety . In this section we discuss the relation between the cohomology complexes of , and . Further details can be found in [57] or [59].
Theorem 2.10**.**
Assume that for all , the order of the group is invertible in (in particular is perfect). Then
[TABLE]
in .
Sketch of proof.
Let be the push-forward of the constant sheaf along the quotient map . Since is finite we have and therefore
[TABLE]
Taking the coinvariants we get, using the projection formula
[TABLE]
It remains to check that the natural map is an isomorphim of sheaves. The stalk of at a point is the permutation module , which is projective by assumption on the order of . Therefore the fact that the previous map is an isomorphism can be checked on the stalks with the usual tensor product. ∎
When is a field and then is a semisimple algebra and complexes of -modules are quasi-isomorphic to their cohomology (see §1.7). Furthermore, invariants and coinvariants are isomorphic as -modules in that case and the previous theorem shows that
[TABLE]
Assume until the end of this section that is either or . Given an -subgroup of and an -permutation module, we denote by the image of the invariants in the coinvariants . It induces an additive functor on the homotopy category of -permutation modules, which we will still denote by . We refer to [63, §27] for basic results on -permutation modules and the Brauer functor.
Theorem 2.11** (Rickard).**
The inclusion induces an isomorphism
[TABLE]
in
representative in (see §2.1)
Sketch of proof.
Assume for simplicity that is a Sylow subgroup of and that . We consider the closed subvariety of defined by
[TABLE]
Then , and hence . Now by Corollary 2.3, the cohomology complex of is perfect therefore its image by the Brauer functor is zero. Using the distinguished triangle in Theorem 2.6 we deduce that which in turn is isomorphic to .
One can generalize this argument to any -subgroup of by considering a filtration of by subvarieties with respect to the size of the -part of the stabilizer of points. ∎
2.4. Trace formula
Assume now that the quasi-projective variety is defined over , and denote by the corresponding Frobenius endomorphism so that in particular . The Frobenius induces a quasi-isomorphism on the complexes and . All the quasi-isomorphisms and triangles listed in §2.2 are compatible with . The corresponding action on the -adic cohomology groups can be computed partially from the number of -points of as follows.
Theorem 2.12** (Lefschetz trace formula [19]).**
[TABLE]
From this theorem we can for example derive a formula for the Euler characteristic of , given by
[TABLE]
Exercise 2.13*.*
Show the latter formula (hint: use the eigenvalues of (and on and Theorem 2.12 for ).
Example 2.14*.*
(a) For the affine space of dimension , we have and . Therefore acts on the cohomology of by multiplication by (the same actually holds for ).
To take the action of into account, we will write , and will be referred to as a Tate twist. With this notation, the -equivariant form of Poincaré-Verdier duality (see Theorem 2.4) is
[TABLE]
for a smooth variety of pure dimension .
(b) and .
(c) and .
Formula (2.1) can be extended to the case of a group action. Assume that the action of on is -equivariant. Then the virtual character of the representation of afforded by the -adic cohomology groups is
[TABLE]
Since this value is both an algebraic integer (left-hand side) and a rational number independent of , this shows in particular that it is an integer independent of . Note however that the individual cohomology groups could depend on , but it was proved recently that this is not the case for Deligne–Lusztig varieties (see [60]).
3. Deligne–Lusztig varieties and their cohomology
This chapter presents the construction by Deligne–Lusztig of algebraic varieties acted on by finite reductive groups [20]. We discuss several properties of the cohomology complexes of these varieties which we will use to deduce representation-theoretic results for finite reductive groups in the following chapters.
Starting from this chapter, will denote a connected reductive algebraic group defined over , and a Frobenius endomorphism defining an -structure on . The group of fixed points is a finite reductive group. Given a closed -stable subgroup of , the corresponding finite group will be denoted by . We refer to Meinolf Geck’s lecture notes [39] for more on these finite groups (see also the textbooks [16, 21]).
The first section of this chapter serves as a motivation for the introduction of Deligne–Lusztig varieties as a generalization of Harish-Chandra induction. It is intentionally very sketchy and will not be used in the rest of these notes.
3.1. Generalizing Harish-Chandra induction
The Harish-Chandra (or parabolic) induction and restriction functors provide an inductive approach to the construction of representations of finite reductive groups. Let be an -stable parabolic subgroup of . It has a Levi decomposition where is an -stable Levi complement and is the unipotent radical of . The finite set is endowed with a left action of by left multiplication, and a right action of by right multiplication (since normalizes ). We can therefore consider the adjoint pair of exact functors
[TABLE]
between the categories and . Howlett–Lehrer showed in [49] that these functors depend only on and not on , up to isomorphism. Therefore they will be simply denoted by and .
There are two issues when working with these functors. The first one is that not every representation occurs in a representation induced from a proper Levi subgroup (unlike the usual induction). The second problem is that an -stable Levi subgroup of in not necessarily a Levi complement of an -stable parabolic subgroup. Even though exists, the finite set might not. However, the variety does and one can consider the following subvariety
[TABLE]
called the parabolic Deligne–Lusztig variety associated with . As in the case of the set , it has a left action of by left multiplication, and a right action of by right multiplication. Consequently, the cohomology complex is a bounded complex of -bimodules and we can consider the triangulated functors
[TABLE]
between the derived categories and . They are called Deligne–Lusztig induction and restriction functors. When (i.e. when is -stable) then is just the finite set and these functors coincide with Harish-Chandra induction and restriction functors.
In these notes we will focus on the case where is a torus. In that case is -conjugate to a quasi-split torus and is determined by an element in . The corresponding variety will be denoted by and studied in the next sections. This case corresponds to the original definition of Deligne–Lusztig [20] which was later generalized to the parabolic case in [50].
3.2. Bruhat decomposition
We fix a pair where is a maximal torus of contained in a Borel subgroup . We assume that both and are -stable (such pairs always exist and form a single -conjugacy class). Such a torus is said to be quasi-split. In that case acts on the Weyl group of .
Given we define
, the length of with respect to ,
the set of simple reflections of with respect to .
Theorem 3.1** (Bruhat decomposition [21, §1]).**
* generates and is a Coxeter system.*
* decomposes as the disjoint union of Bruhat cells*
[TABLE]
\mathbb{B}s\mathbb{B}w\mathbb{B}=\left\{\hskip-3.69885pt\begin{array}[]{l}\mathbb{B}sw\mathbb{B}\text{ if }\ell(sw)>\ell(w)\\ \mathbb{B}w\mathbb{B}\sqcup\mathbb{B}sw\mathbb{B}\text{ otherwise.}\end{array}\right.**
The Schubert cell is isomorphic to (the affine space of dimension ).
Note that coincides with the length corresponding to the Coxeter system . Indeed, , therefore the inequality forces by (iii). More generally is the smallest integer such that with .
The closure in of a Bruhat cell is a closed subvariety of stable by left and right multiplication by . Therefore by (ii) it must be a finite union of Bruhat cells. We consider a partial order on , called the Bruhat order, defined by if (or equivalently ). Then by (ii) we have
[TABLE]
The singularities of these varieties are of considerable interest for the study of representations of semisimple Lie algebras and reductive groups, not only for finite reductive groups.
Example 3.2*.*
(a) For the trivial element of we have , which is a closed subvariety of . Therefore is the unique minimal element for the Bruhat order.
(b) The variety is irreducible, therefore there exists a unique such that . The element is the unique element of maximal length in , and its length equals the dimension of the flag variety , which is the number of positive roots of . The element is also the unique maximal element for the Bruhat order.
Example 3.3*.*
For we have , hence
[TABLE]
3.3. Deligne–Lusztig varieties
Let be the unipotent radical of the Borel subgroup . We fix a set of representatives of in . The Deligne–Lusztig varieties attached to are
[TABLE]
The finite group acts by left multiplication on both and . Furthermore, acts by right multiplication on . Indeed, if then using that normalizes we have, for every
[TABLE]
Now if and only if which we can rewrite as .
Using properties of Schubert cells, one can prove that both and are smooth quasi-projective varieties of pure dimension . Furthermore, the canonical projection induces a -equivariant isomorphism
[TABLE]
As in the previous chapter we can consider the cohomology complexes attached to these varieties. The complex is a bounded complex of finitely generated -bimodules, and is a bounded complex of finitely generated -modules. Since acts freely on , we deduce from Theorem 2.10 and (3.1) that
[TABLE]
Example 3.4*.*
If then . The latter is just a finite set isomorphic to . Similarly, and therefore the cohomology complexes of and are given by a single permutation module in degree [math], namely
[TABLE]
Example 3.5*.*
Let and be the standard Frobenius of , raising the entries of a matrix to the th power, so that . The usual subgroups of can be chosen as follows:
[TABLE]
Then the varieties and are given explicitly by
[TABLE]
and the cosets and by
[TABLE]
Finally, the element is given by
[TABLE]
We deduce the following explicit descriptions of the varieties and .
[TABLE]
The variety is the famous Drinfeld curve, discovered and studied by Drinfeld in [23].
Since , the algebra is semisimple and the complex is quasi-isomophic to its cohomology as a complex of -bimodules. Given an irreducible character of , we can consider the -isotypic part of each individual cohomology groups . Since the actions of and commute, is a -module. Given we set
[TABLE]
The function is the character of the virtual module (or equivalently of the complex ). It is called a Deligne–Lusztig character. A particular interesting case is when . We have
[TABLE]
since in that case the isotypic component of corresponding to the trivial representation of is the invariant part under which by (3.2) is isomorphic to .
3.4. Properties of and
In the examples 3.4 and 3.5 the varieties and are affine. This was proven in general by Deligne and Lusztig [20] when (the Coxeter number of ) and it is conjectured to hold unconditionaly (see for example [7] for further examples). In any case, the consequences on the vanishing of the cohomology groups (see §2.1) hold.
Theorem 3.6** (Lusztig [52]).**
* for .*
Consequently, the complexes and can be represented by complexes with (-free) terms in degrees . In addition, one can compute the stabilizer of any point in under the action of and invoke Corollary 2.3 to show the following additional properties.
Proposition 3.7**.**
The complex is perfect. Furthermore, if the order of is invertible in then is perfect as well.
More generally, one can show that if in an -subgroup of such that , then is necessarily conjugate to a diagonal subgroup of .
Example 3.8*.*
Let and \dot{w}=\dot{s}=\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right) so that
[TABLE]
If (see Example 3.5) then if and only if is an eigenvector of with eigenvalue . In particular, either is unipotent (with eigenvalues ) or is conjugate to .
If is invertible in then there is no non-trivial -subgroup such that . In particular is perfect as a complex of bimodules in that case. Otherwise we have the following result (see [1, §9] for basic results on vertices and sources).
Proposition 3.9**.**
Let be a representative of as a complex of -permutation modules with no null-homotopic direct summand. Then the vertices of the terms of are contained in .
Proof.
Let be an -subgroup of which is not conjugate to a subgroup of . Then and therefore by Theorem 2.11 we have .
Now assume that there is an indecomposable direct summand in such that . Without loss of generality we can assume that is maximal for this property. Then for any other direct summand of the terms of , is either zero or projective. Consequently is an acyclic complex with projective terms. Take to be in the largest degree of so that for and . Then is a split surjection (by [4, Lem. A.1]), which contradicts the minimality of . ∎
3.5. Applications
The following result will be intensively used in the rest of these notes. It was first proved by Lusztig in the case where [52] and then extended by Bonnafé–Rouquier to the modular setting.
Theorem 3.10** (Bonnafé–Rouquier [5]).**
Let be a simple -module and be minimal for the Bruhat order such that R\mathrm{Hom}_{\Lambda G}\big{(}R\Gamma_{c}(\mathbb{Y}(\dot{w}),\Lambda),M\big{)}\neq 0. Then there exists a representative of such that
each is a finitely generated projective -module (in degree ),
* is a direct summand of for only (middle degree).*
In other words, the terms of that complex satisfy .
Sketch of proof.
The key property shown by Bonnafé–Rouquier is that the cone of the natural map lies in the thick subcategory of generated by the complexes for . In particular, the minimality of show that this map induces an isomorphism
[TABLE]
terms in degrees
terms in degrees
Consequently, the cohomology of vanishes outside the degree . In other words, for .
Now let be a representative of with projective terms and with no null-homotopic direct summand. Let be maximal for the property that is a direct summand of , and assume that . If the composition is zero, then the map induces a morphism between the complexes and . Therefore it must be null-homotopic by the above argument, which is impossible since is not a direct summand of . Therefore the composition must be non-zero, hence surjective, which shows that is a (null-homotopic) direct summand of . By assumption of , this is again impossible. ∎
Corollary 3.11** (Bonnafé–Rouquier [5]).**
The triangulated category of perfect complexes is generated by the cohomology complexes for .
Proof.
We show by induction on the length of that for every simple -module , if then lies in the thick subcategory of generated by the complexes for . This is true for since (by definition a thick subcategory is stable under direct summands).
Let , and consider a minimal representative of as a bounded complex of projective modules. It follows from Theorem 3.10 that the indecomposable direct summands of for already appear in the cohomology complexes for . By induction lies in the thick subcategory of generated by the complexes for . But the term in middle degree can be written as
[TABLE]
which proves that lies in the category generated by for .
To conclude, it remains to show that any projective indecomposable module appears as a direct summand of a minimal representative of for some . It is enough to show it at the level of characters. This follows from the fact that the regular representation of is uniform, which means that it is a linear combination of Deligne–Lusztig characters . ∎
One can invoke Corollary 3.11 to see that a morphism of bounded complexes of -modules is a quasi-isomorphism if and only if for all . This was a key step in Bonnafé–Rouquier’s proof of the Jordan decomposition as a Morita equivalence (see [5]).
The analogue of Corollary 3.11 for general bounded complexes (whose terms can have non-trivial vertices) was proved recently by Bonnafé–Dat–Rouquier in [4]. This again was proven essential to show that the Jordan decomposition is a splendid Rickard equivalence. Recall that denotes the (unique up to homotopy equivalence) representative of as a complex of -permutation modules (see §2.1).
We say that the prime number is very good for if is good for every simple component of and for every component of of type . A sufficient condition for to be very good is where is the Coxeter number of .
Theorem 3.12** (Bonnafé–Dat–Rouquier [4]).**
Assume that is very good. Let be the set of complexes where runs over the -subgroups of , and . Then
The thick subcategory of generated by coincide with , where is the additive category generated by the indecomposable modules with one-dimensional sources and abelian vertices.
The image of in generates as a triangulated category closed under direct summands.
4. Decomposition numbers from Deligne–Lusztig characters
Recall that denotes an -modular system with being a finite extension of . We will furthermore assume from this chapter on that and are big enough for all the groups considered (all the group algebras over and will split).
The purpose of this chapter is to explain how one can compute -decomposition numbers for finite reductive groups using the Deligne–Lusztig characters introduced in the previous chapter. We start by recalling elementary results on decomposition matrices (see for example [61, §14]) before explaining the case of finite reductive groups.
4.1. Grothendieck groups and dualities
Let be an abelian (resp. additive) category. The Grothendieck group of , denoted by (or sometimes ) is the abelian group generated by the isomorphism classes of objects of subject to the relations for every short exact sequence (resp. split short exact sequence) . Note that any abelian category is additive. When there is a risk of confusion, the Grothendieck group of as an additive category will be referred to as the split Grothendieck group. If is an abelian category in which every object has finite composition length, then can be identified with the free abelian group with basis . Similarly, the split Grothendieck group of a Krull-Schmidt category is the free abelian group generated by the isomorphism classes of the indecomposable objects.
Let be a triangulated category. The Grothendieck group of is the abelian group generated by the isomorphism classes of objects of subject to the relations for every distinguished triangle .
Given an abelian category , the fully faithful functor induces an isomorphism . Under this identification, the class of a bounded complex is
[TABLE]
Similarly, for any additive category the functor induces an isomorphism between the corresponding Grothendieck groups.
The Grothendieck groups we will be interested in in this chapter are: -basis given by simple modules
basis given by projective indecomposable modules (PIMs)
Since exact sequences split in and then for , in (resp. ) if and only if . This is not true in general in since -modules can have non-trivial extensions when divides the order of (which is the interesting case from our point of view).
In addition, there are perfect pairings
[TABLE]
defined by when and are actual modules and is the field or . Then is a self-dual basis for the pairing , whereas a dual basis of for is given by the classes of projective indecomposable modules.
4.2. Lifting projective modules
We say that a -module lifts to characteristic zero if there exists an -lattice (an -module which is free as an -module) such that as -modules. Not every -module can be lifted to characteristic zero in general, unless in which case is semisimple. This holds nevertheless for projective modules. Indeed, given a finitely generated projective -module , we can consider the projective cover of as an -module. Then and the -module is an -module lifting . In addition, it is projective and it is up to isomorphism the unique projective -module lifting . Note that if is a simple -module, then , together with the composition is a projective cover of , viewed as a simple -module.
Given and its lift to characteristic zero, we can form the -module . Its character (or rather its image in the Grothendieck group) will be denoted . This defines a group homomorphism
[TABLE]
4.3. Decomposing ordinary characters
Let be a -module. One can choose an -free -submodule such that . By saturating by the action of one can assume that is stable by , so that is an -lattice such that . Then one can form the -module and consider its image in , which we will denote by .
Proposition 4.1**.**
* is well-defined and extends to a group homomorphism*
[TABLE]
called the decomposition map.
(Brauer reciprocity)* is the transpose of for the pairings defined in §4.1. In other words*
[TABLE]
Proof.
It is enough to prove (ii). Let be a -module and be a projective -module. We constructed -lattices and such that and . Then
[TABLE]
∎
The decomposition matrix (or ) is the matrix with entries
[TABLE]
for and . With this notation we have
[TABLE]
Example 4.2*.*
(a) If then every simple -module is projective, and lifts to an irreducible ordinary character. Consequently is the identity matrix up to reordering.
(b) If is an -group, then the only irreducible -module is the trivial representation. Since the decomposition map preserves the dimension, we deduce that for every irreducible ordinary character of . The decomposition matrix in that case is a column encoding the dimensions of the irreducible -modules.
Dually, the projective cover of the trivial representation is the regular representation , which lifts to characteristic zero as , and whose character is . This is an example of Brauer reciprocity as stated in Proposition 4.1.ii.
(c) Let us consider the particular case of and . There are three irreducible representations over : the trivial representation , the sign and the reflection representation, of dimension . The latter has an integral version given by The representations and yield two non-isomorphic representations over by -reduction, the trivial representation and the sign modulo , which we still denote by . There is a short exact sequence
[TABLE]
which shows that . We deduce that the decomposition matrix in that case is
[TABLE]
Consequently, the two PIMs have characters and .
4.4. Basic sets of characters
We mentioned in a previous section that not every -modules can be lifted to characteristic zero. This is however true at the level of the Grothendieck groups: the class of any -module is the -reduction of a virtual character.
Theorem 4.3**.**
The decomposition map is surjective.
By Proposition 4.1, the map is the transpose of , hence it is injective. Therefore if and are two projective -modules then if and only if their lifts and have the same character. In other words, a projective module is determined by its character over .
Since is surjective, it is natural to search for a set of ordinary irreducible characters such that is a -basis of . Such a set is called a basic set (see [40]). If it exists, then the decomposition matrix has the following shape:
[TABLE]
In that case, a projective -module is determined by the projection of its character on .
Now assume that is a finite reductive group. Recall that the unipotent characters are the irreducible constituents of the virtual characters
[TABLE]
for . The unipotent blocks are the -blocks containing at least one unipotent character, and the irreducible characters in the union of unipotent blocks are the constituents of
[TABLE]
for and an irreducible -character of .
Exercise 4.4*.*
Given , show that in .
Recall that is said to be very good for if is good for every simple component of and for every component of of type . A sufficient condition for to be very good is where is the Coxeter number of .
Theorem 4.5** (Geck–Hiss, Geck [40, 38]).**
Assume that is very good. Then the unipotent characters form a basic set for the union of unipotent blocks.
More generally, under the same assumption on , given a semisimple -element, the Lusztig series is a basic set for the union of blocks where runs over the set of semisimple -elements of .
4.5. Decomposition numbers and Deligne–Lusztig characters
This section is the core of this chapter and contains recent results on -decomposition numbers for unipotent blocks when is not too small (see for example [25, 28]). We start by listing the different tools and assumptions we are going to use to determine these numbers.
(HC) The Harish-Chandra restriction/induction of a projective -module remains projective (this follows easily from the biadjointness and the exactness of the functors).
(Uni) When is very good, the restriction of the decomposition matrix to the set of unipotent characters (a basic set by Theorem 4.5), ordered by increasing -function, has unitriangular shape. This is only conjectural (see [41, Conj. 3.4]).
(Hecke) The decomposition matrix of the Hecke algebra (corresponding to the unipotent principal series) embeds in the decomposition matrix of the finite group .
(Reg) If is an ordinary irreducible -character of in general position (i.e. is irreducible) then
[TABLE]
for every projective -module . This gives a non-trivial information since is only a virtual unipotent character, even though is irreducible.
Proof.
Using the fact that when (see Exercise 4.4) and Brauer reciprocity we have
[TABLE]
Since is assumed to be irreducible, the sign of this scalar product coincides with . ∎
Recall from Proposition 3.7 that the complex is perfect as a complex of -modules. Let denote its class in . Then
[TABLE]
The following property is a character-theoretic consequence of Bonnafé–Rouquier’s theorem 3.10. It is particularly suited for determining decomposition numbers on cuspidal -modules, as we will see in the examples of the next section.
(DL) Given a simple -module , let be minimal (for the Bruhat order) such that (i.e. occurs in ). Then .
Proof.
Let be a simple -module and be such that . Take to be a reduced representative of with each projective. By assumption, there exists such that is a direct summand of . Taking to be maximal, we deduce that
[TABLE]
By Theorem 3.10 if is minimal for the property that the complex is non-zero then has a representative such that occurs only in middle degree. Therefore and cannot be minimal for the property that . ∎
4.6. Examples in small rank
We discuss here three examples of small-rank finite reductive groups where the previous tools allow a complete determination of the (unipotent part of the) decomposition matrix. Here the assumption (Uni) was shown to hold by Geck [35] for finite linear and unitary groups and by White [64] for .
(a) We start with . Its order is . Assume that is an odd prime number with and . Then the principal block contains the two unipotent characters and , together with some non-unipotent characters. Using (Uni) and (Hecke) we have
[TABLE]
Here the value of can be obtained by (Hecke). Indeed, if then is semisimple and with a simple -module. Therefore in this case. Now if then is a PIM with character , hence .
(b) Let , whose order is . We will denote by and the simple reflections in corresponding to the short and long simple root respectively. Assume in this example that is odd and . Then the characters in the principal block are
[TABLE]
in the principal series
cuspidal
There is another unipotent character, denoted by , which under the assumptions on is of defect zero and this forms a block by itself. We have , giving all the unipotent characters lying in the principal series. The first approximation of the decomposition matrix is given by (Uni)
[TABLE]
A (Hecke) argument shows that is indecomposable. Since it is projective, it gives the first column of the decomposition matrix.
The second and third column can be obtained by (HC). First, let and be the PIM of such that non-unipotent characters (see Example (a)). Let be the block idempotent corresponding to the principal block. Then is projective and its character, cut by the block, is given by
[TABLE]
Similarly with the Levi subgroup we get
[TABLE]
Consequently the unipotent part of the decomposition matrix is
[TABLE]
with .
We use (Reg) to determine the exact value of and . Since there exists a non-trivial -character . Furthermore, if then one can choose to be lying outside the reflection hyperplanes (in the reflection representation of on the group of characters of ). In that case it is in general position, and (Reg) yields for every projective -module . Now , so if we apply this to the projective indecomposable modules , and corresponding to the second, third and fourth columns of the decomposition matrix we get
[TABLE]
which gives (and hence ) and .
The final ingredient is (DL). To use it we decompose each virtual projective module on the basis of PIMs. To this end, recall from (4.1) that . We have
[TABLE]
Since we deduce from (DL) that must be a non-negative combination of and , since and do not appear in the decomposition of for . Writing
[TABLE]
we deduce that , which forces . We conclude that the unipotent part of the -decomposition matrix (when is odd and ) is given by
[TABLE]
Exercise 4.6*.*
We follow the notation of [16, §13] for unipotent characters. Complete the determination of the -decomposition matrix of when and , which is given by
[TABLE]
with (see [45]). To this end, use (DL) with the following values of the Deligne–Lusztig characters, cut by the principal block ,
[TABLE]
Here and denote the simple reflections in the Weyl group of type corresponding to the short and long simple root respectively.
(c) Let “”. Its order is given by
largest defect
[TABLE]
We will work again in the case where . In addition we will assume that to ensure the existence of -characters in regular position. As in the case of linear groups, the unipotent characters of are parametrized by partitions of . Here, they are and are all contained in the principal -block. By (Uni) the unipotent part of the decomposition matrix has the following shape
[TABLE]
By (Hecke) the projective -module decomposes as a direct sum of two PIMs. The corresponding decomposition of characters is which gives the first and third columns of the decomposition matrix.
As in the previous examples, other columns can be determined by Harish-Chandra induction of projective modules of various Levi subgroups. The Levi subgroup of type has two interesting PIMs and with respective characters
[TABLE]
yielding by (HC) two projective -modules and with characters
[TABLE]
Note that these projective modules might not be indecomposable.
If there exists an -character of in general position and (Reg) applies. In other words, for every projective -module . With
[TABLE]
we have . Therefore for every direct summand of and . From this we deduce that
is indecomposable,
with and indecomposable. But is impossible (the PIM cannot lie in the Harish-Chandra series of both and ).
This gives the second and fourth column of the decomposition matrix. As in the previous example, application of (Reg) to the fifth and sixth column of the decomposition matrix gives lower bounds for the decomposition numbers and has the following shape
[TABLE]
The two missing columns correspond to projective covers of cuspidal simple -modules. They can be obtained using (DL) from the decomposition of each on the basis of PIMs. The minimal representatives of the -conjugacy classes of are ordered as follows under the Bruhat order: . Let , , be the PIMs ordered as the columns of the decomposition matrix. Then
[TABLE]
Note that in the virtual module , the module appears with negative multiplicity. This is consistent with (DL) which ensures that since it does not occur in , it must occur with a multiplicity whose sign is given by .
[TABLE]
Since , we must have and . But recall that which we can rewrite as . This forces , hence (otherwise would be negative) and therefore . Consequently and , , have yet to occur.
[TABLE]
which, since is even forces and therefore . We obtain finally
[TABLE]
4.7. Example in
The methods described in §4.6 and used in the example of in the previous section have shown powerful to determine completely the decomposition matrices for small-rank groups, up to the decomposition matrix of (see for example [25, 28, 29]). We give here a general example of a decomposition number that can be computed using Deligne–Lusztig characters. For the sake of simplicity we have chosen again the case of a group of type , although the proof can be adapted to other classical groups.
Theorem 4.7** (Dudas–Malle [28]).**
Assume that , and . Then .
Proof.
Let be an irreducible -character of in general position (such a character exists since ). Then is an irreducible character, and hence
[TABLE]
with each . Among the simple -modules such that , choose with smallest possible such that . In other words, if then for every simple -module .
If is any simple -module such that , then by minimality of and (DL) we have . In particular, for every simple -module . Let us write
[TABLE]
By the orthogonality relation of Deligne–Lusztig characters, this sum is zero unless . Since all the terms are non-negative, and one term is positive (for ) we deduce that . In other words, if occurs in some for then . Now
[TABLE]
Therefore the -submodule of spanned by the virtual projective modules for has codimension . We deduce that there is at most one simple -module such that . It must be since and
[TABLE]
This proves that .
Now (Uni) shows that the character of the projective cover of is of the form . Using the fact that it is orthogonal to we get
[TABLE]
which shows that
[TABLE]
Indeed, the scalar product is, up to a sign, equal to the value on of the irreducible character of corresponding to the partition . So here, it equals the dimension of the representation of corresponding to which is by the hook length formula. ∎
4.8. Observations and conjectures
The following conjectures were made by Geck in [34] and Geck–Hiss in [41].
Conjecture 4.8** (Geck–Hiss).**
Assume that and that is large with respect to .
The decomposition matrix has a unitriangular shape.
If is unipotent and cuspidal then is irreducible i.e. for some simple -module .
The unipotent part of the decomposition matrix is independent of (it depends only on the order of in ).
The conjecture was first proven for groups of type and by Geck [35] and for classical groups when is linear by Gruber–Hiss [44]. In addition, all the decomposition matrices computed so far for small rank group satisfy the conjecture. In a recent preprint [30], Malle and the author proved the part (ii) of the conjecture under the extra assumption that is good.
The difficulty in proving this conjecture lies in producing nice projective modules. In the case where the conjecture is known to hold, generalized Gelfand–Graev modules were used. Malle and the author proposed a different strategy in [27], using again the cohomology of Deligne–Lusztig varieties.
Conjecture 4.9** (Dudas–Malle).**
For all there is a sign such that
[TABLE]
Alvis–Curtis duality
intersection cohomology
is the character of a (non-virtual) projective module.
The ’s are actually virtual projective modules whose character can be explicitly computed using Kazhdan–Lusztig polynomials. They satisfy a unitriangularity property (as we expect for the PIMs) and the multiplicities on unipotent characters do not depend on .
Example 4.10*.*
For the variety is dense in and hence . It is a smooth projective variety paved by affine spaces , therefore
[TABLE]
Consequently, . There is indeed a PIM whose character is , hence is the character of a projective module as claimed in Conjecture 4.9.
5. Brauer trees of unipotent blocks
This chapter is devoted to the study of unipotent blocks of finite reductive groups with cyclic defect groups. In that case the structure of the block is encoded in a planar embedded tree, the Brauer tree. We explain how to use the cohomology complexes of Deligne–Lusztig varieties to get information on the characters of PIMs (which gives the tree as a graph) and extensions between simple modules (which gives the planar embedding). This is based on a recent work of Craven, Rouquier and the author [17].
5.1. Brauer trees
Throughout this chapter, is any finite group and is an -block of with cyclic (and non-trivial) defect groups. We denote by a defect of . The results on the structure of originate in a work of Brauer [9] subsequently completed by Dade [18] and Green [43]. For a self-contained treatment of blocks with cyclic defect groups we recommend [58] and [1, §V].
We denote by the set of irreducible ordinary characters of lying in . There is a set called the set of exceptional characters of such that if we define then the character of any projective indecomposable -module in is given by
[TABLE]
with and . In other words, a simple -module occurs in the -reduction of either two distinct non-exceptional characters, or in one non-exceptional character and in every exceptional character.
We define the Brauer graph of as the graph with vertices labeled by and edges for every PIM such that . The edges of the Brauer graph are therefore labeled by PIMs or equivalently by simple -modules (via their projective cover). The knowledge of the Brauer graph, together with the multiplicity of the exceptional vertex, is equivalent to the knowledge of the decomposition matrix.
Example 5.1*.*
(a) Let be the cyclic group of order . There is only one PIM which is the projective cover of the trivial module. One can write its character as
[TABLE]
The convention here is to define the exceptional characters as the non-trivial characters of . This gives the Brauer graph given in Figure 1.
(b) Let us now consider the case of where is an -subgroup of . In particular the order of divides . As in the previous example is indecomposable and hence it forms a single block.
Every simple -module has a trivial action of the -group , and therefore it must be of the form . In addition, since is an -group, the restriction is semisimple hence simple. We deduce that the simple -modules are in bijection with the irreducible representations of (over or ). More precisely, if we fix a generator of and a primitive -th root of unity in , one can consider the simple -modules to be the one-dimensional representation of on which acts by . They lift to characteristic zero to one-dimensional -modules and -modules on which acts by . Let . Then:
The -modules for form a set of representatives of .
lifts to characteristic zero as . We denote by the character of .
The projective cover of is , which lifts to characteristic zero as . It has character
[TABLE]
where denotes the sum of the irreducible characters of which are non-trivial on . These are the exceptional characters of .
We deduce that the Brauer graph is a star-shaped tree as shown in Figure 2.
(c) Let and be the principal -block (with defect a Sylow subgroup of ). The irreducible ordinary characters of are labeled by partitions of . The characters in correspond to partitions which are -hooks
[TABLE]
Here denotes the sign characters. The subgroup of is an -group, therefore every irreducible character is the character of a projective module. Consequently, the same holds for the induction of any representation from to . Using the branching rules for induction we have
[TABLE]
projective character
not in
This shows that the Brauer graph is a line as shown in Figure 3. Note that here any vertex can be chosen to be the exceptional vertex.
Theorem 5.2** ([1, §23]).**
The Brauer graph is a tree, called the Brauer tree of .
A planar embedding of is defined by an ordering of the set of edges incident to any given vertex. Planar embedded trees will be drawn according to the anti-clockwise order around a vertex.
Theorem 5.3** (Structure of PIMs [1, §22]).**
Let be the Brauer tree of .
There exists a unique ordering around each vertex of such that if and are two simple -modules labeling edges incident to a given vertex then follows immediately if and only if .
Given a simple -module labeling an edge between non-exceptional vertices as follows
\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\psellipticarclinestyle=dotted,linewidth=1.5pt(2,2)193287 \psellipticarc[linewidth=1pt]-¿(2,2.5)(1,1)205275 \psellipticarclinestyle=dotted,linewidth=1.5pt(2,2)13107 \psellipticarc[linewidth=1pt]-¿(6,2.5)(1,1)2595
the Loewy structure of is given by
[TABLE]
Together with this planar embedding, the Brauer tree will be referred to as the planar embedded Brauer tree of . This tree encodes the structure of the block . Indeed, if is a block of with cyclic defect groups, then and are Morita equivalent if and only if the planar embedded Brauer trees of and coincide, and the multiplicity of the exceptional vertices are equal.
Remark 5.4*.*
Note that the structure of can also be described in the case where the edge of is connected to the exceptional node. In that case one needs to repeat the composition series a number of times equal to the multiplicity of the exceptional vertex. In other words, one needs to turn around the exceptional vertex times and consider that all the other vertices have multiplicity one.
The proof of the structure theorem consists in constructing a stable equivalence between the block of and its Brauer correspondent in . This equivalence is built from the Green correspondence between and its Brauer correspondent in and a Morita equivalence between the latter block and its Brauer correspondent in (which is ). For a block with normal defect group (as in the case of ), the Brauer tree is a star, as shown in Example 5.1.b, and the structure theorem is easily proved. For more details see for example [58] or [1, §V].
The structure theorem for the PIMs has the following consequence, which we will use in the following section. Let be an irreducible ordinary character of which labels a leaf of (a vertex with only one incident edge). The -reduction of is the simple -module which labels the unique edge incident to that leaf. This is the particular case of Theorem 5.3 where . Then is a uniserial module with composition factors . Moreover, it lifts to an ordinary character labeling the unique vertex adjacent to the leaf, say , so that . Similarly, is a uniserial module which lifts to an ordinary character such that . The edge labeled by is the edge which comes directly after in the cyclic ordering around . If we iterate this process, we see that can be obtained by following the edges in a walk around the tree. More precisely, we obtain a sequence of ordinary characters and simple -modules such that:
is a uniserial module with head and socle .
lifts to an -lattice with character (which is either or irreducible, and hence labels a vertex).
.
labels the edge coming directly after in the cyclic ordering around the vertex labeled by .
If denotes the number of isomorphism classes of simple -modules, then . Each simple -module appears exactly twice in the sequence , called the Green walk around [43].
Example 5.5*.*
We consider the planar embedded tree as shown in Figure 4. Then for example lifts to a lattice with character whereas is uniserial with composition series , and it lifts to a lattice with character . The sequence is , whereas the sequence is .
5.2. The case of unipotent blocks
We now focus on the case of finite reductive groups. As before, is a connected reductive group over together with a Frobenius endomorphism defining an -structure. Throughout this section and until the end of this chapter we will assume for simplicity that is split (i.e. acts trivially on the Weyl group).
We denote by the order of modulo , or equivalently the order of the class of in . The integer is minimal for the property that , where is the -th cyclotomic polynomial. Recall that when is very good, the unipotent -blocks are “generic” and parametrized by -cuspidal pairs.
[TABLE]
When is cyclic, the non-exceptional characters in are the unipotent characters in , which are the irreducible constituents of the virtual character
[TABLE]
where for some parabolic subgroup with Levi complement . The variety is the parabolic Deligne–Lusztig variety attached to (see §3.1). Recall that when is a maximal torus of type and is conjugate to the Borel subgroup by then . In that case the non-exceptional characters in the block are the constituents of the Deligne–Lusztig character .
We first list the algebraic methods which can be used to determine the Brauer trees of unipotent blocks of with cyclic defect groups. The first three arguments are not specific to finite reductive groups, whereas the last one relies on results by Geck [37].
(Parity) If then modulo (the dimension of a projective module is divisible by ).
(Real stem) If is real, the real characters form a single connected line containing the exceptional node called the real stem of . The complex conjugation induces a symmetry of with respect to that line.
(Degree) The dimension of a non-exceptional character equals the sum of the dimensions of the simple modules labeling the edges incident to in .
(Hecke) The Brauer trees of the blocks of the Hecke algebra associated to a given Harish-Chandra series in (as defined by Geck in [37]) are subtrees of . Each of these subtrees is a line, with dimension (or rather -function) increasing towards the exceptional vertex.
Example 5.6*.*
Let be a finite reductive group of type . We denote by and the two simple reflections of its Weyl group . The degrees of are and , therefore the order of the finite group is
[TABLE]
In this decomposition the exponents of the cyclotomic polynomials and are equal to . Therefore when (when is good) and divides one of these polynomials, the Sylow -subgroups of are cyclic and the principal block has cyclic defect groups.
Assume that and , in which case has order modulo (the Coxeter number). A torus of type is a -Sylow subgroup since . The trivial character of this torus is -cuspidal, and the corresponding block via (5.1) is the principal block, whose characters are
[TABLE]
since . As in §4.6, we use the notation of [16, §13] for the unipotent characters. In particular is a primitive third root of in . The only non-real unipotent characters are and , therefore the real stem is
\nbput[npos=1.1]\vphantom{\big{)}}G_{2}[-1]\ncput[npos=1.135]\naput[npos=1.1]\vphantom{\big{)}}\mathrm{St}\ncput[npos=1.135]\naput[npos=1.1]\vphantom{\big{)}}\phi_{2,1}\ncput[npos=1.13]\naput[npos=1.15]\vphantom{\big{)}}1\ncput[npos=1.13]\psbracenodesepA=-2.1cm,nodesepB=13pt,rot=90,braceWidthInner=5pt,braceWidthOuter=0.05pt,braceWidth=0.5pt(8.15,1.1)Brauer tree of the Hecke algebra
By (Parity) the complex conjugate characters and must be connected to either the vertex labeled by or the exceptional vertex. But if . Now would force , which is a prime number that we excluded. Therefore (Degree) forces to be as in Figure 5.
Exercise 5.7*.*
If then , and is a -Sylow subgroup for . Then the non-exceptional characters of the principal block, with their parity, are
[TABLE]
Here the only non-real characters are again the complex conjugate characters and . Show that in that case the Brauer tree is given as in Figure 6.
Theorem 5.8**.**
The Brauer trees of unipotent blocks are known for
* of classical type , , and (Fong–Srinivasan [32, 33]).*
* of exceptional type except and .*
Note that for groups of small rank, the determination of the trees follows from the determination of all the -decomposition matrices for various , which were more specifically solved by Burkhart [14] for , Shamash [62] for , Geck [36] for , Hiss [46] for and , and Wings [65] for . The determination of the other trees were obtained by Hiss–Lübeck [47] for and , and Hiss–Lübeck–Malle [48] for . In addition to the algebraic arguments used in the example of , it is often necessary to use partial information on the character table of the group to determine the tree (in order to decompose tensor products of characters).
The problem for larger exceptional groups such as and comes from the increasing number of cuspidal -modules. As in the case of decomposition matrices (see Chapter 4), these representations resist to algebraic methods which rely on Harish-Chandra induction and restriction, such as (Hecke). The idea developped in [17] by Craven–Rouquier and the author is to use the cohomology complexes of Deligne–Lusztig varieties to produce perfect complexes satisfying the following proposition.
Proposition 5.9**.**
Let be a perfect complex of -modules. Assume that there exist integers such that for . Then
[TABLE]
Consequently, the -modules and differ only by their projective summands.
Proof.
Using (1.2) we get the following distinguished triangle in
[TABLE]
which we can also write as
[TABLE]
Now by Theorem 1.24 the image in is also a distinguished triangle. Since is perfect, its image in the stable category is zero. Thus we obtain the following distinguished triangle in
[TABLE]
which yields in . ∎
Given , we can consider the complex obtained from the cohomology complex of by cutting by the generalized -eigenspace of , and by the block . In other words, the complex is isomorphic to a direct summand of such that . The requirements to use Proposition 5.9 are
- (1)
must be perfect. This follows from Proposition 3.7 if we assume .
- (2)
The condition on the vanishing of should already hold over by the universal coefficient formula (see §6.3). This supposes a vanishing of many of the groups for every eigenvalue congruent to modulo .
- (3)
The vanishing over should follow from the vanishing over and the property that each cohomology group is -free.
In the few examples where has been explicitly computed, it is torsion-free whenever (in other words (3) follows from (1)). The reader will find an example of this property in the following chapter, for varieties associated with Coxeter elements.
Example 5.10*.*
Let be a finite reductive group of type . We denote by the simple reflections in . If has order modulo , in which case divises then the principal -block has cyclic defect groups. It corresponds to the cuspidal pair where is a element of of order (such an element can be taken to have length ). The fourteen non-exceptional characters in the block are given by the irreducible constituents of the Deligne–Lusztig character . They consist of eight unipotent characters in the principal series (including and ), four unipotent characters in the -series and the two cuspidal unipotent characters of , namely the complex conjugate characters and . The real stem of the Brauer tree is formed by the real characters (all except ) with increasing dimension towards the exceptional vertex by (Hecke), see Figure 7.
The missing characters in the tree are the complex conjugate characters . Unlike the case of in Example 5.6, a combination of the (Degree) and (Parity) arguments is not enough to determine to which vertex they are attached. To remove the ambiguity, we consider the Deligne–Lusztig variety attached to a Coxeter element and the corresponding cohomology complex. Here we will consider the generalized eigenspaces of corresponding to the eigenvalues and . Over , the cohomology of , cut by the block, is
[TABLE]
with the convention that modulo (here in ). We obtain
[TABLE]
If we assume that is torsion-free, then the universal coefficient theorem shows that has only one non-zero cohomology group. By using the truncation functors of Proposition 1.19 it follows that it is quasi-isomorphic to a single projective -module in degree , which lifts to a lattice of character . This shows that the vertex labeled by is connected to the vertex labeled by and gives the Brauer tree of . Even better, using Proposition 5.9 with the complex which has only two non-zero cohomology groups yields
[TABLE]
in the stable category . This proves that is an -lattice with character and the planar embedded Brauer tree shown in Figure 8 is obtained from the Green walk.
Exercise 5.11*.*
Show that the perfect complex is homotopy equivalent to
[TABLE]
This method has proven very powerful in the case of exceptional groups of type and , removing also some ambiguity in the planar embedding of the Ree groups . Luckily, only small-dimensional Deligne–Lusztig varieties were needed to complete the determination of the trees, which made checking the vanishing of the cohomology possible.
Theorem 5.12** (Craven–Dudas–Rouquier [17]).**
All the planar embedded Brauer trees of unipotent blocks of finite reductive groups are explicitly known in terms of Lusztig’s parametrization of unipotent characters.
6. The Coxeter variety
The last chapter of these notes is devoted to the study of Deligne–Lusztig varieties attached to a special class of Weyl group elements, called the Coxeter elements. These varieties were first studied by Lusztig [51]. Computing their -adic cohomology turned out to be a key ingredient in Lusztig’s subsequent work on the classification of ordinary irreducible characters of finite reductive groups. We explain in this chapter how to extend Lusztig’s result to the modular setting, building on work of Bonnafé–Rouquier [6] and the author [24, 26].
Throughout this chapter we will assume for simplicity that is split (i.e. acts trivially on the Weyl group). All the main results of this chapter can be generalized to the case where a power of acts trivially (which includes the case of the Ree and Suzuki groups).
6.1. Geometry of the Coxeter variety
Let be a split maximal torus of , contained in an -stable Borel subgroup of . Recall that the simple reflections of the Weyl group are exactly the elements such that has dimension .
A Coxeter element of is a product of all the simple reflections, in any order. Coxeter elements are conjugate under . The order of any Coxeter element is the Coxeter number of , denoted by . A Coxeter variety is a Deligne–Lusztig variety or attached to a Coxeter element. Since Coxeter elements are the elements of minimal length in their conjugacy class, they are actually conjugate by a sequence of cyclic shifts. Consequently, the cohomology of a Coxeter variety does not depend on the choice of a Coxeter element. For that reason we shall denote these varieties simply by or . For more details on Coxeter elements see [8, §V.6] and for Coxeter varieties see [51].
Example 6.1*.*
Let . The permutation is a Coxeter element of . It has length and order . For the general linear group the flag variety can be identified with the set of flags of vectors spaces such that . With this description, the Deligne–Lusztig variety attached to an element is
[TABLE]
Recall that two flags and are said to be in relative position if there exist such that is a basis of and is a basis of for each . With we deduce that if and only if is a basis of and is a basis of . This can be written , which yields in turn
[TABLE]
This gives an explicit description of the Coxeter variety in the case of as
[TABLE]
If then is obtained from by removing some simple reflections. Therefore it is a product of simple reflections lying in a proper subset of , and as such it is a Coxeter element of the parabolic subgroup of . We shall write . Consequently,
[TABLE]
Let (resp. ) be the standard parabolic subgroup of (resp. standard Levi subgroup of ) attached to . Its unipotent radical will be denoted by . We will write for the Coxeter variety of . We have and therefore
[TABLE]
There are two key properties of the Coxeter variety that are needed to compute its cohomology (and to show that it is torsion-free). The first one is given by (6.1). The second is a result of Lusztig [51] giving the quotient of by unipotent subgroups in terms of Coxeter varieties of Levi subgroups.
Proposition 6.2**.**
Let . There is a (non-equivariant) isomorphism of varieties
[TABLE]
which induces an -equivariant isomorphism of -adic cohomology groups
[TABLE]
In particular, with being the empty set we deduce that . One can actually refine Lusztig’s result as follows (see [26, Lem. 4.2]): if is the complement of in then
[TABLE]
Again, this isomorphism is not equivariant for the action of or in general.
6.2. Cohomology over
Using a combination of (6.1), Proposition 6.2 and computations of for (in order to use Lefschetz trace formula, see Theorem 2.12) Lusztig gave in [51] a complete description of the cohomology of over , with the action of and .
Theorem 6.3** (Lusztig).**
A cuspidal character occuring in the cohomology of occurs in the middle degree only.
The eigenspaces of on give mutually non-isomorphic irreducible representations of .
The eigenvalues of on , restricted to a given Harish-Chandra series, are of the form
[TABLE]
for some root of unity and some . No constituent of for lie in that Harish-Chandra series.
Sketch of proof.
If then for some proper subset of and from (6.2) we get H_{c}^{\bullet}(\mathbb{X}(v),K)\simeq R_{L_{I}}^{G}\big{(}H_{c}^{\bullet}(\mathbb{X}_{I},K)\big{)}. Therefore contains no cuspidal character. By Theorem 3.10, it follows that cannot be a constituent of for , which proves (i). In addition, one can show using the purity of that has eigenvalue with on the -isotypic part of . Furthermore, it is a general property of the cohomology of Deligne–Lusztig varieties that is actually a root of unity.
Let be a cuspidal character of for some and set . The eigenvalue of on the -isotypic part of is of the form for some root of unity . If is an irreducible character of lying in the Harish-Chandra series of then is a non-zero multiple of . In particular, the eigenvalues of on the Harish-Chandra series of correspond to the eigenvalues of on the -isotypic part of {}^{*}R_{L_{I}}^{G}\big{(}H_{c}^{\bullet}(\mathbb{X},K)\big{)}, which by Proposition 6.2 and (i) are times the eigenvalues of on the cohomology of . Assertion (iii) follows.
The proof of (ii) requires to compute the endomorphism algebra of , which would go beyond the scope of these notes. ∎
Example 6.4*.*
(a) Let . Recall that the unipotent characters of are parametrized by partitions of . We will represent them by their Young diagram. Then the cohomology of the Coxeter variety (given in Example 6.1) with the eigenvalues of is
[TABLE]
(b) Let a group of type . Using the notation in [16, §13] for the unipotent characters of (in particular is a primitive third root of and a primitive fourth root of ), the cohomology of is given by
[TABLE]
cuspidal characters
series above
6.3. Cohomology over
Since is not flat over , the mod- cohomology of a variety is not the -reduction of the cohomology over , but there is still an explicit relation, called the universal coefficient theorem, given by the following exact sequence (see for example [2, §2.7])
[TABLE]
We will use it in the following particular case, for which we can give a direct proof.
Proposition 6.5**.**
The middle cohomology group of the Coxeter variety is torsion-free.
Proof.
We consider an -free resolution of which we truncate using (see §1.6 for the definition of the truncation). This yields quasi-isomorphisms
[TABLE]
where each is -free. Furthermore, since the map is injective (note that is injective by definition). On the other hand, since and therefore is -free the exact sequence tensored with yields an exact sequence
[TABLE]
which forces . ∎
Theorem 6.6**.**
Assume that . Then is torsion-free.
Proof.
We proceed by induction on the semisimple rank of . If is a torus, then is a point and the result holds. Otherwise, if is non-empty, one can consider a proper subset of . Using the isomorphism of varieties given in Proposition 6.2 we get an isomorphism of -modules
[TABLE]
Note that we do not assume this isomorphism to be -equivariant. By induction, the cohomology of (a Coxeter variety for the Levi subgroup ) is torsion-free. This shows that the torsion part of is killed under Harish-Chandra restriction, and hence it is cuspidal.
By the universal coefficient formula, a cuspidal -submodule of yields a subquotient of . Let be the largest degree of which has a cuspidal subquotient (or equivalently since is semisimple, a direct summand). If then Theorem 3.10 forces the -module to occur in the cohomology of a Deligne–Lusztig variety for . But this is impossible by (6.2). This, together with Proposition 6.5, shows that the cohomology of is free over . ∎
Question. Does the result hold for other Deligne–Lusztig varieties ? If so, can we replace the condition by ?
From now on we assume that divides , the -th cyclotomic polynomial evaluated at . We will also assume that so that is actually the order of modulo . We first observe from the explicit values of the eigenvalues of given in [51, Table 7.3] that:
The classes in of the eigenvalues of on are exactly the -th roots of unity in .
Under the assumption on , all the proper standard Levi subgroups of are -groups. Consequently, the proof of Theorem 6.6 shows that
The torsion-part of is a cuspidal -module.
Following Theorem 6.3.iii, let (resp. ) be the smallest (resp. largest) eigenvalue of on within the Harish-Chandra series corresponding to (resp. to ). By Theorems 6.3 and 6.6 for , together with (i) we deduce the following property
Let be a proper subset of . Then the generalized eigenspaces of on the cohomology of for the eigenvalues and satisfy and for .
Using this observation, we get, for
[TABLE]
by (iii)
by Poincaré duality
by (iii)
which is zero since . This proves that
[TABLE]
In addition, the universal coefficient formula shows that is the mod- reduction of the -module . This information can be used in combination with the following result, which holds for a more general class of Deligne–Lusztig varieties.
Theorem 6.7** (Dudas–Rouquier [31]).**
Let and be the class of in . Then
[TABLE]
in .
Idea of proof.
One can compute explicitly the closed subvariety of consisting of the points such that divides the order of . Then the cohomology complexes of and are isomorphic in . ∎
Choose such that modulo . Then Theorem 6.7 and (6.4) show that up to projective summands. This proves that lifts to an -lattice with character , yielding information on the Green walk around the Brauer tree of the principal -block when .
Example 6.8*.*
Let be a group of type , so that . Assume that has order modulo . We choose (resp. ) to be congruent to (resp. ) modulo . The various data attached to the representations occurring in the cohomology group of in middle degree are listed in the following table.
[TABLE]
We get therefore , , , and the planar embedded Brauer tree is given in Figure 9.
Knowing only the characters of the PIMs (in other words, the Brauer tree without the planar embedding) we can finally show that the cohomology of over is torsion-free when .
Theorem 6.9**.**
Assume and , so that has order modulo . Then is torsion-free.
Idea of proof.
Since the torsion part of the cohomology of over is cuspidal (see (ii) above), it is enough to show that for every simple cuspidal -module , the complex has zero cohomology outside the middle degree . Indeed, by the universal coefficient formula this shows that the torsion-part of is in degree only, and we can invoke Proposition 6.5 to conclude.
The computation of is achieved by using the explicit character of as follows. The shape of the tree, as conjectured in [48] and proved in [31] ensures that labels an edge incident to the exceptional vertex. In other words, we have .
We distinguish two cases : assume first that is cuspidal, then does not occur in any of the cohomology complexes for since otherwise would occur in (recall that is torsion-free). Consequently, the map induces an isomorphism
[TABLE]
which proves that the cohomology of this complex vanishes in degrees higher than and lower than .
Assume now that lies in a Harish-Chandra series above a proper Levi subgroup of . Writing , one shows that is a direct summand of for some (projective) -module . Now, with (resp. ) being a lattice lifting , we obtain the following isomorphism using adjunction and (6.3)
[TABLE]
The cohomology of this complex is torsion-free by Theorem 6.6. Therefore the same holds for . Now its cohomology over vanishes outside of the degree since occurs in the cohomology of in middle degree only, and by the universal coefficient theorem the same holds over . ∎
6.4. Applications
Broué’s abelian defect group conjecture [10] predicts the existence of a derived equivalence between a block of a finite group with abelian defect and its Brauer correspondent. In the case of finite reductive groups, defect groups of unipotent blocks are generic. When is large enough and is the order of modulo , they correspond to the -part of -tori in , and their centralizers are -Levi subgroups (see §5.2). Broué suggested in [10] that in this case the derived equivalence should be induced by the cohomology complex of a Deligne–Lusztig variety associated with such a -Levi subgroup. This was proven by Bonnafé–Rouquier and the author in the case when is the Coxeter number.
Theorem 6.10** (Bonnafé–Rouquier [6], Dudas [24]).**
Assume that has order modulo . The action of on can be extended to an action of such that the functor
[TABLE]
induces a derived equivalence between the principal blocks of and .
The extension of the action of to is essentially given by twisting the action of the Frobenius endomorphism, since here is a cyclic group generated by the image of . For more general blocks, it is expected that the braid operators constructed in [13, 22] should provide the extension of the action of the finite torus (see [12]).
Once we extended the action, the key point is to prove that is a tilting complex, that is that
[TABLE]
Rouquier proved in [59] that it is enough to show the vanishing of the cohomology groups of that complex over , that is to show that
[TABLE]
The solution to this problem given in [7] and [24, 26] relies on showing first that the cohomology groups are torsion-free and then to use the cohomology over (computed by Lusztig) to determine and find an explicit representative for .
Proposition 6.11**.**
Let be the unipotent character corresponding to the generalized -eigenspace of on for some . With the following notation for the subtree of the Brauer tree of the principal -block of corresponding to the Harish-Chandra series of
\ncput*[npos=0.5]\ncput*[npos=0.5]\ncput*[npos=0.5]\naput[npos=1.1]\vphantom{\big{)}}\chi_{\lambda}
the complex is isomorphic to
[TABLE]
Sketch of proof.
Lusztig’s result on the quotient of (Proposition 6.2) can be generalized to as follows. Let us decompose the torus as as a product of an -group and an -group. We define . It is an intermediate quotient between and whose cohomology contains only the principal -block of . Then one shows that for every proper subset of there is an isomorphism of -modules
[TABLE]
In particular, the torsion part of the cohomology of is cuspidal. As in the case of , it is enough to show that cuspidal modules occur in the middle degree only. This was proven for along the way to Theorem 6.9. The same property holds for since is built from successive extensions of in the same way that is built from extensions of the trivial representation.
By definition, the cohomology groups are the sum of the cohomology groups where runs over the irreducible -characters of . When , which we know explicitly. When is non-trivial, the assumption on forces to be in general position and except when in which case it equals the exceptional character . Consequently, a given eigenspace of on in non-zero in at most two degrees, one corresponding to the eigenspace on , and the other being the middle degree. Since is a direct summand of the perfect complex (recall that it corresponds to the principal block of ) then is also perfect and therefore is the character of a virtual projective module. This forces each generalized -eigenspace of to be of the following form
[TABLE]
where is the unique degree of the cohomology of on which acts by and is the corresponding unipotent character (we assume here that is one of the eigenvalues listed in Theorem 6.3.iii).
From the shape of the Brauer tree we observe that is exactly the distance between the node labeling and the node labeling . On the other hand, since the cohomology of is torsion-free we deduce from (6.6) and Proposition 5.9 that
[TABLE]
in the stable category. Both cohomology groups lift to -lattices with characters and , therefore they have no projective summands and the previous isomorphism holds in .
Recall that we have a distinguished triangle in
[TABLE]
which is determined by a map , which is in turn determined by an element of . By Proposition 1.23 and the isomorphism (6.7), this group of extensions is isomorphic to and hence it is one-dimensional. Therefore up to isomorphism there is a unique non-zero map , and the mapping cone of this map can be obtained from a truncated projective resolution of , which is exactly the complex given in the theorem. ∎
Now this representative is exactly the one given by Rickard in [56] to construct a tilting complex for Brauer trees algebras. In particular, it satisfies (6.5) and Theorem 6.10 follows.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] J.L. Alperin , Local representation theory , Cambridge Studies in Advanced Mathematics 11 , Cambridge University Press, Cambridge, 1986.
- 2[2] D.J. Benson , Representations and cohomology, Vol.1: Basic representation theory of finite groups and associative algebras , Cambridge Studies in Advanced Mathematics 30 , Cambridge University Press, Cambridge, 1991.
- 3[3] C. Bonnafé , Representations of SL 2 ( 𝔽 q ) subscript SL 2 subscript 𝔽 𝑞 \mathrm{SL}_{2}(\mathbb{F}_{q}) , Algebra and Applications 13 , Springer-Verlag London, Ltd., London, 2011.
- 4[4] C. Bonnafé, J.-F.Dat and R. Rouquier , Derived categories and Deligne-Lusztig varieties II, Ann. Math. 185 (2017), 609–670.
- 5[5] C. Bonnafé and R. Rouquier , Catégories dérivées et variétés de Deligne–Lusztig, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 1–59.
- 6[6] C. Bonnafé and R. Rouquier , Coxeter orbits and modular representations, Nagoya Math. J. 183 (2006), 1–34.
- 7[7] C. Bonnafé and R. Rouquier , Affineness of Deligne-Lusztig varieties for minimal length elements, J. Algebra 320 (2008), 1200–1206.
- 8[8] N. Bourbaki , Lie groups and Lie algebras , Chapters 4–6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002.